Question

The rate constant is given by the equation \(\mathrm{K}=\mathrm{P} . \mathrm{Ze}^{-\mathrm{EKT}}\). Which factor should register a decrease for the reaction to proceed more rapidly (a) \(\mathrm{T}\) (b) \(\mathrm{Z}\) (c) \(\underline{\mathrm{E}}\) (d) \(\mathrm{P}\)

Short answer

Decrease \( \underline{E} \) to increase the reaction rate.

Step-by-step solution

Identifying Reaction Rate Constants

The rate constant, \( K \), is influenced by several factors in the equation \( K = P \cdot Z \cdot e^{-\frac{E}{KT}} \). This equation shows how pressure (\( P \)), collision frequency (\( Z \)), activation energy (\( E \)), and temperature (\( T \)) impact the rate.

Understanding Exponential Decay

The term \( e^{-\frac{E}{KT}} \) indicates an exponential decay behavior with respect to \( E \) and \( T \). If \( E \), the activation energy, decreases, the exponential term increases because the negative exponent becomes less negative, thus increasing \( K \).

Analyzing Factors for Rate Increase

For the reaction to proceed more rapidly, \( K \) should be increased. Hence, amongst the given factors, reducing \( E \) would increase the exponent \( e^{-\frac{E}{KT}} \), thus increasing \( K \) and speeding up the reaction.

Key concepts

Rate Constant

The rate constant, often denoted by the symbol \( K \), is a fundamental component in chemical kinetics. It serves as a proportionality factor in the rate equation of a chemical reaction. This constant indicates how quickly a reaction proceeds. Its value is affected by various factors, including the nature of the reactants and conditions such as temperature. The formula for the rate constant typically includes terms for pressure, collision frequency, and activation energy, as in the equation \( K = P \cdot Z \cdot e^{-\frac{E}{KT}} \). Here, \( P \) stands for pressure, \( Z \) represents the collision frequency of the molecules, \( E \) is the activation energy needed for the reaction to occur, and \( T \) is the temperature. Each of these factors plays a role in determining the rate constant:

• Pressure \( P \): Changes in pressure can influence how often molecules collide, therefore affecting \( K \).
• Collision frequency \( Z \): Indicates how often reacting molecules collide, an essential component for reactions to take place.
• Activation energy \( E \): The minimum energy required for a reaction to proceed. Lower \( E \) implies more molecules can react, thus increasing \( K \).
• Temperature \( T \): Typically an increase in temperature results in an increase in \( K \), as molecules move faster and collide more frequently.

Reaction Kinetics

Reaction kinetics is the study of the rates at which chemical reactions occur and the factors that affect these rates. It provides insight into the steps and interactions between molecules that lead to the formation of products. Understanding these kinetics is crucial for applications in developing new chemical processes and improving existing ones. In the context of the exercise, reaction kinetics explores how the variables in the rate constant equation \( K = P \cdot Z \cdot e^{-\frac{E}{KT}} \) interact to influence the rate of a reaction. The rate at which a reaction proceeds can be altered by one or multiple components of this equation. Key concepts in reaction kinetics include:

• Reaction order: Reflects how the rate is affected by the concentration of reactants.
• Rate laws: Mathematical relationships that describe how the rate depends on reactant concentrations.
• Complex reactions: Occasionally, reactions involve multiple steps and intermediates, complicating the kinetic analysis.

By manipulating conditions such as temperature or pressure, or altering catalysts used in the reaction, scientists can optimize reaction rates for desired outcomes.

Collision Theory

Collision theory provides a framework for understanding how particles interact to result in a chemical reaction. According to the theory, molecules must collide with sufficient energy and proper orientation for a reaction to occur. This theory underscores the importance of the activation energy \( E \) in determining reaction rates. Central to collision theory is the idea that not all collisions result in successful reactions. Only those collisions that possess enough energy to overcome the activation energy barrier and that occur with the correct alignment will succeed in forming products. Key points about collision theory include:

• Collision frequency \( Z \): Refers to the number of collisions per unit time in a system, contributing significantly to reaction rates.
• Energy threshold: Only collisions with energy equal to or greater than the activation energy \( E \) can result in a reaction.
• Steric factor or orientation effect: Collisions must occur with the reactants oriented correctly to allow for product formation.

This theory helps to explain why increasing temperature typically boosts reaction rates, as higher temperatures increase particle velocity, leading to more frequent and energetic collisions.